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5  Let $n\ge 1$ be an integer and let us work over the field of complex numbers. Let $\mathcal{R}_n$ denote the set of rational conic bundles $\pi\colon X\to \mathbb{P}^n$ (morphisms such that the generic fibre is a geometry irreducible conic and $X$ is rational), up to square equivalence: two conic...

3  I posted this question on MSE a few days ago, but got no response (despite a bounty). I hope it will get more answers here, but I'm afraid it might not be appropriate as I'm not sure it's actually research-level. Please do tell me if it's not appropriate and if possible tell me how to modify the ... Conic bundle
In algebraic geometry, a conic bundle is an algebraic variety that appears as a solution of a Cartesian equation of the form X 2 + a X Y + b Y 2 = P ( T ) . {\displaystyle X^{2}+aXY+bY^{2}=P(T).\,} Theoretically, it can be considered as a Severi–Brauer surface, or more precisely as a Châtelet surface. This can be a double covering of a...  Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology on algebraic varieties, introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring a topological space, called the spectrum of the ring. The Zariski topology allows using tools of topology for the study of algebraic varieties, even when the underlying field is not a topological field. This is one of the basic ideas of scheme theory, which allows one to build general algebraic varieties by gluing together affine varieties in a way similar to that in manifold...